\(n\)  being an integer (\(n\) =0 for axial symmetry), \(k\)  the real wavelength of the instability, and w an acceptable frequency solving the equations in plane \(z\) =constant with \(v_1\) =0 on cylinders with radii \(r=R_1\)  and \(r=R_2\)  (see also the paper on triskels by [1]}. The Reynolds number is evaluated by \(\Omega _1 R_1^2 /\nu \)  (or \(\Omega _2 R_2^2 /\nu \) ). For fixed \(n\)  and \(k\) , the \(\omega \)  become a discrete set of \(\omega _n(k)\)  values. In the case of the Earth, \(v_1\) =0 at the solid surface and at the atmosphere free surface, and the only observed solutions are 24 h and 1 yr. We therefore expect to find harmonics of these two values, The detailed study for two concentric cylinders can be found in [2]}. The \(v_1\)  flow is stationary and consists in toric vortices (or Taylor vortices) regularly spaced along the cylinder generatrices. If the series of periods \(\omega _n(k)\)  linked to the annual oscillation is 1, 1/2 1/3, 1/4 yr (see section 3), then one expects to find experimentally perturbations of order \(n\) =3 or 4, as was already the case for trends ([3]}; Figure 8 top). We find in the present study that indeed such is the case for the annual (Figure 7 middle) and semi-annual (Figure 7 bottom) oscillations.
